1. Field of the Invention
The present invention relates to a deformation measuring method and apparatus using electronic speckle pattern interferometry; and, more specifically, to a deformation measuring method and apparatus using electronic speckle interferometry which can favorably perform phase unwrapping when analyzing temporal deformation of a dynamic object.
2. Description of the Prior Art
For measuring the surface shape and deformation of an object in a mirror surface condition by interferometry, demands for simply determining its phase distribution (corresponding to the surface shape) have been becoming very strong especially in the optical and electronic fields due to the advancement of technology in recent years. As techniques for determining the phase distribution of such an object to be observed in interferometric measurement in particular, those mainly using phase shifting and spatial Fourier transform (Fourier transform introducing therein a spatial carrier; ditto in the following) have conventionally been known.
In general, the phase-shifting method determines a phase distribution of an object according to pattern information of respective interference pattern images, whose phases are shifted from each other by a phase angle which is an integral fraction of 2π, between object light and reference light in an interferometer. The intensity signal in a predetermined spatial domain obtained from N interference pattern images equally dividing 2π by N is given by the following expression (1):Ii(x,y)=I0[1+γ cos {φ(x,y)+2π/N}]i=0,1, . . . ,(N−1)  (1)where φ(x,y) is the phase to be determined. On the other hand, I0 is the average intensity, and γ is the visibility (modulation) of interference pattern, both of which are unknown quantities. In a simple example where N=4, φ(x,y) is determined by the following expression (2):                               ϕ          ⁡                      (                          x              ,              y                        )                          =                              tan                          -              1                                ⁡                      (                                                            I                  2                                -                                  I                  4                                                                              I                  1                                -                                  I                  3                                                      )                                              (        2        )            
The Fourier transform method will now be explained. When one of optical paths of an interferometer is tilted by a minute angle θ about the y axis, a spatial intensity distribution given by the following expression (3) is obtained:                                                                         I                ⁡                                  (                                      x                    ,                    y                                    )                                            =                            ⁢                                                I                  0                                ⁡                                  [                                      1                    +                                          γcos                      ⁢                                              {                                                                              ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                +                                                      2                            ⁢                                                                                                                   ⁢                            π                            ⁢                                                                                                                   ⁢                            x                            ⁢                                                                                                                   ⁢                            sin                            ⁢                                                                                                                   ⁢                                                          θ                              /                              λ                                                                                                      }                                                                              ]                                                                                                                        =                                ⁢                                                      I                    0                                    ⁡                                      [                                          1                      +                                              γcos                        ⁢                                                  {                                                                                    ϕ                              ⁡                                                              (                                                                  x                                  ,                                  y                                                                )                                                                                      +                                                          2                              ⁢                                                                                                                           ⁢                              π                              ⁢                                                                                                                           ⁢                                                              f                                0                                                            ⁢                                                                                                                           ⁢                              x                                                                                }                                                                                      ]                                                              ⁢                                                                                                       (        3        )            where f0=sin θ/λ is the spatial frequency of carrier fringes generated by tilting the optical path. When the above-mentioned expression (3) is Fourier-transformed in the x direction, the following expression (4):                                                                         J                ⁡                                  (                                      f                    ,                    y                                    )                                            =                            ⁢                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                      I                    ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                                      exp                    ⁡                                          (                                                                        -                          j                                                ⁢                                                                                                   ⁢                        2                        ⁢                                                                                                   ⁢                        π                        ⁢                                                                                                   ⁢                        f                        ⁢                                                                                                   ⁢                        x                                            )                                                        ⁢                                                                           ⁢                                      ⅆ                    x                                                                                                                          =                            ⁢                                                                    J                    0                                    ⁡                                      (                                          f                      ,                      y                                        )                                                  +                                                      J                    1                                    ⁡                                      (                                                                  f                        +                                                  f                          0                                                                    ,                      y                                        )                                                  +                                                      J                    1                    *                                    ⁡                                      (                                                                  f                        -                                                  f                          0                                                                    ,                      y                                        )                                                                                                          (        4        )            is obtained.
The right side of expression (4) indicates that the three terms can be separated from one another on the spatial frequency axis if f0 is sufficiently large. When f0 is made sufficiently large so as to take out the second term of the right side alone, whereas the other terms are cut by a filter, and the second term is shifted to the origin of the spatial frequency domain the following expression (5):Ir=I0γ exp[iφ(x,y)]  (5)is obtained.
From the ratio (arctangent) between the real part and imaginary part of the right side of expression (5), the following expression (6)                               ϕ          ⁡                      (                          x              ,              y                        )                          =                              tan                          -              1                                ⁢                                    Im              ⁢                                                           [                              I                r                            ]                                      Re              ⁢                                                           [                              I                r                            ]                                                          (        6        )            is obtained.
In each of the two conventional methods mentioned above, the unknown quantity φ(x,y) to be determined is obtained independently of the other unknown quantities I0 and γ. The phase value is determined as a principal value between [−π, π]. FIG. 8 shows how the phase is obtained according to expression (6) when the object phase linearly changes. The denominator and numerator in the above-mentioned expression (6) are cosine and sine functions, respectively, and the arctangent of their ratio yields a saw-tooth phase distribution in which phase hopping occurs at intervals of 2π. The phase-hopped positions are determined, and the phase value of 2π is added to (or subtracted from) data on the right side of each of these positions, so as to correct the phase hopping, whereby a phase distribution in proportion to the object shape can be obtained. This processing is known as phase unwrapping.
In the phase-shifting method, the phase of a point in a two-dimensional interference pattern can be analyzed from several intensity data obtained by phase shifting of this point in general. Therefore, this point will not be influenced by other points spatially different therefrom. In the Fourier transform method, by contrast, all the data on a line constituted by spatial carriers must be subjected to Fourier transform operations, whereby the phase of a point cannot be determined independently from other points.
On the other hand, the object to be observed must stand still throughout a period in which several phase-shifted interference pattern images are captured in the phase-shifting method. By contrast, the Fourier transform method is considered to be suitable for dynamic phenomena, since phase analysis can be carried out from a single interference pattern image only if a carrier component is made beforehand. In other words, the phase-shifting method requires the object to be temporally constant though spatial restrictions thereon are loose, whereas the Fourier transform method makes it necessary for the object to have a phase distribution which is spatially moderate enough as compared with the spatial carrier period or uniform but allows the object to move temporally.
However, these methods are not applicable to cases where the object temporally changes while being spatially nonuniform, or changes drastically. For example, the process from plastic deformation of a material to destruction thereof is nonlinear, so that a temporal and spatial deformation distribution must be measured when determining distortions thereof. However, the above-mentioned two methods are theoretically hard to apply to such measurement.
Dynamic speckle pattern interferometry has been known as an interferometry method effective in a case where object has a rough surface, and such a temporal and spatial fluctuation exists.
Speckle pattern interferometry is an interferometric method utilizing a freckle-like pattern (speckle pattern) occurring in the observation surface of a rough object illuminated by laser light. In typical imaging systems, the speckle pattern is considered unfavorable as image noise. However, it carries phase information, so that deformation can be estimated from changes in the phase information. In addition, the speckle pattern interferometry enables highly accurate deformation measurement with reference to the wavelength of light.
FIG. 9 shows a speckle interferometer of a dual illumination type. An object 100 to be observed, which is a rough surface object, is illuminated with two luminous fluxes 102A, 102B from a laser source 101 arranged substantially symmetrical to each other within the x-z plane. The light fields scattered by the object 100 form an interference speckle pattern on the imaging surface of a CCD camera 103. Thereafter, thus obtained interference speckle pattern image is analyzed, whereby phase analysis is carried out to obtain the surface shape of the object 100.
As the phase analysis technique, subtraction-addition method has been known.
In the phase analysis of speckle pattern images in general, respective speckle patterns before and after the object 100 is deformed are captured, and differences in intensities of corresponding image points therebetween are calculated. The differential intensity approaches zero at places with a stronger correlation, i.e., where the phase change caused by deformation is 0 or an integral multiple of 2π, whereas a greater value of differential intensity is obtained at places with a weaker correlation, whereby a correlation pattern corresponding to the amount of deformation can be obtained if the absolute value of intensity difference Isub between the two images is calculated.
In the above-mentioned subtraction-addition method, information of intensity sum Iadd is utilized in addition to information of intensity difference Isub, whereby a phase is determined independently from the visibility (modulation) γ.
Here, as shown in FIG. 9, shutters 104A, 104B are placed on the respective optical paths of two luminous fluxes 102A, 102B, so as to measure intensity distributions I1(x;t), I2(x;t) of individual image in a temporal domain when the object is illuminated by only one of the luminous fluxes beforehand.
In general, an interference pattern I(x;t) obtained in an optical system (in-plane deformation or out-of-plane deformation system) of a speckle interferometer is represented as in the following expression (7):I(x;t)=I0(x;t)[1+γ(x;t)cos(θ(x;t)+φ(x;t))]  (7)where I0(x;t) is the average intensity of I1(x;t) and I2(x;t), θ(x;t) is a speckle random phase, γ(x;t) is the modulation, and φ(x;t) is the object phase.
Subsequently, letting Ibefore (t=t1) be the intensity before object deformation, and Iafter (t=t2) be the intensity after object deformation, they are respectively represented as the following expressions (8) and (9):Ibefore=I(x;t1)=I0[1+γm cos(θ+φ1)]  (8)Iafter=I(x;t2)=I0[1+γm cos(θ+φ2)]  (9)
The difference Isub and sum Iadd of these intensity patterns are calculated, while local averages are determined at the same time, as represented by the following expressions (10) and (11):
 Isub=<|Iafter−Ibefore|>=c<|sin(θ+Δφ/2)|><|sin(Δφ/2)|>≈c′|sin(Δφ/2|  (10)Iadd=<|Iafter+Ibefore−2I0|>=c<|cos(θ+Δφ/2)|><|cos(Δφ/2)|>≈c′|cos(Δφ/2)|  (11)where < > indicates a local spatial average, whereas c and c′ are constants. The average intensity I0 can be determined by a local temporal average. Δφ=φ2−φ1 and indicates the deformation of object between the time t1 and time t2.
The object phase can be determined if the above-mentioned two components are subjected to the arithmetic operation represented by the following expression (12):                               Δ          ⁢                                           ⁢          ϕ                =                  2          ⁢                                           ⁢                      tan                          -              1                                ⁢                                    I              sub                                      I              add                                                          (        12        )            
When the above-mentioned subtraction-addition method is used, however, the absolute value is taken for each of sine and cosine components as represented by the expressions (10) and (11), whereby the phase curve actually obtained for an object phase linearly increasing as shown in FIG. 10 has phase folding points as indicated by a solid line in FIG. 10 though not generating the above-mentioned phase hopping. For this matter, each folding section may be referred as numeral n, so that phases φ are connected (phase-unwrapped) by using the following expression (13):                                           ϕ            ′                    =                    ⁢                                    2              ⁢                                                           ⁢              π              ⁢                                                           ⁢              int              ⁢                                                           ⁢                              (                                  n                  2                                )                                      +                                                            (                                      -                    1                                    )                                                  n                  -                  1                                            ⁢              ϕ                                      ⁢                                  ⁢                              n            =            1                    ,          2          ,          3          ,          4          ,                                           ⁢          …                                    (        13        )            whereby the original object phase φ′ can be restored.
In typical interferometric methods, no absolute values are employed for the sine and cosine components used in the arithmetic operation (e.g., the above-mentioned expression (6)) corresponding to the above-mentioned expression (12) for determining the object phase, whereby the resulting phase exhibits saw-tooth phase hopping as shown in FIG. 11 (or FIG. 8). Since clear discontinuities occur with a phase jump of 2π, a phase connecting operation for restoring the object phase can be executed relatively easily in this case. Namely, it would be more useful if discontinuities are automatically detected (by software), and an offset of an integral multiple of 2π is added to or subtracted from this section. In the above-mentioned subtraction-addition method, however, peculiar folding of phase occurs as shown in FIG. 10. As a consequence, no discontinuities occur, whereby it is not easy for software to process folded sections automatically. In general, speckle noise is remarkable in speckle pattern interferometry as shown in FIG. 12. In regions where speckle correlation fringes become dense, i.e., where the amount of deformation is large, the phase connecting operation in the subtraction-addition method becomes difficult in particular, whereby automatic processing is likely to impossible in many cases.
Also, as can be seen from the above-mentioned expressions (10) and (11), it is fundamentally essential for the subtraction-addition method to carry out a local spatial average, that degrades the spatial resolution. This makes it further difficult to detect phase folding points, whereby the phase connecting operation becomes quite difficult.
In view of such difficulty in the phase connecting operation of the subtraction-addition method, the inventors proposed an improvement (Japanese Unexamined Patent Publication No. 2001-311613) over this technique, which has raised the accuracy in detecting phase folding points, but has not been able to detect these points automatically yet.
Meanwhile, attention has recently been given to temporal Fourier transform introducing a temporal carrier unlike spatial Fourier transform introducing a spatial carrier. If the temporal Fourier transform method is applied to the above-mentioned speckle pattern interferometry using a signal having a relatively large noise component, however, the above-mentioned signal will be hard to process by automatically determining an optimal band-pass filter therefor.